Range-Renewal Structure Of Transient Simple Random Walk On Discrete Group

Given a discrete time stochastic process, Rn is a set of visited points during the first n steps.The process that Rn evolution with time n,we call it Range-Renewal process.Early research in this area is done by P.Erdos,A.Dvoretzky,S.J.Taylor and so on.They do some resesrch on range-renewal structure of simple symmetric random walk on Zd.They mainly use analysis methods to estimate the mathematical expecta-tion and variance of some variables. But these methods are complicated for calculation.However, in this paper.we use the method of ergodic theory,which greatly simplifies the complicated calculation process and gives us a unified method for transient simple random walk on discrete group. This paper mainly studies range-renewal structure of transient simple random walk on discrete group.Our results are as follows:Here, Rn (respecitvely Rn,k,Rn,k+) is the number of distinct cites visited at least once (resp. exactly k times, at least k times) by the random walk up to time n and γ is the escape rate of the random walk.